Economic activity at one location generates economic activity at nearby locations. Generally the generated activity declines with the distance from the original location. This paper investigates some mathematical model of this geographic dispersion.

One Dimensional Model

Let the economy be laid out along a line from −R to +R. The level of density of some autonomous activity such as production in industries in the economic base is given by f(r), where r is the distance from the midpoint of the linear economy. It is assumed that a unit of economic activity at r gives rise to a level of dispersed secondary activity of λ. The level of economic activity at s due to the econonmic activity at r is equal to λh(s-r), where h(z) is some known function that sums to unity. Then the level of economic activity a location s, y(s) could be given by

y(s) = f(s) + ∫_−R^R(y(s)λh(s−r))dr
where
∫_−R^Rh(z))dz = 1.

This is a linear integral equation. It has a solution for known values of the function h(z). If the functions over the range [−R, +R] are represented as vectors then the equation takes the form

Y = F + λHY
and thus has the solution
Y = (I−λH)⁻¹F

where H is a matrix and I is the identity matrix.

The above model needs to be adjusted for boundary effects. For a location near the boundary if dispersed activity cannot go beyond the boundary then the effects at the other locations are scaled upwards. In terms of the previous model this involves dividing by the term ∫_−R^Rh(s−r)dr. Thus

y(s) = f(s) + [∫_−R^R(y(s)h(s−r))dr]/∫_−R^Rh(s−r)dr]

In terms of the matrix version the correction would be subsumed in terms of premultiplication of the matrix H by a diagonal matrix D whose diagonal elements are the reciprocals of ∫_−R^Rh(s−r)dr.

Thus

Y = F + λDHY
and thus has the solution
Y = (I−lambda;DH)⁻¹F

In effect, there is a multiplier of (I−λDH)⁻¹ which operates upon the exogenously located economic activity.

The Two Dimensional Model

It would be very messy to specify the model as an integral equation for the two dimensional case but in terms analogous to vectors it is manageable. Let F(r,θ) and Y(r, θ) be matrices given in terms of polar coordinates. The equation for the two dimensional case would be the same as for the one dimensional case except the definitions of multiplication would be changed.